Search Results

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the …

This note contains an affirmative answer to the question https://maco.lu.ac.ir/article-1-86-en.html

Is there a polynomial vector field on $\mathbb{R}^2$ whose corresponding singular complex foliation of $\mathbb{C}^2 $ admits a complex limit cycle $L$ such that $L$ does not intersect the real pl …

Is there a polynomial vector field $X$ on $\mathbb{R}^2$ such that every complex limit cycle of the corresponding foliation of $\mathbb{C}^2$ must necessarily intersect the real plane $\mathbb{R}^2$. …

Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties? The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ t …

This paper contains a conjecture and a partial result about the abelian integral under discussion :"Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals" …

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{ …

Inspired by the two posts which are linked bellow we ask the following question: Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ wi …

Motivations: We first introduce our motivations: We wish to find an operator-theoretical interpretation for the number of limit cycles of a polynomial vector field on the plane. We quote the stateme …

Inspired by this question and the counter example provided in its answer we ask: Is there a polynomial vector field on $\mathbb{R}^2$ such that after complexification of the equation, the cor …

A revision: Novembre 2020 I am realy indebted to Loic Teyssier for his $2$ very valuable comments and suggestions. I summarize his comments as follows: To have a hyperbolic complex limit cycle …

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in thi …

What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related questio …

First we explain our Motivation: Motivation: First note that there is no a Riemannian metric on an open set of the plane which possess two nested closed geodesics $\gamma_1, \gamma_2$ …